16.6 Parametric Surfaces and Their Areas
Claudia Castro-Castro
Math 283 Spring 2020
Example: Graphs of functions \( z=f(x,y) \)
- Let \[ x(u,v) =u, \qquad y(u,v) =v,\qquad z(u,v)=f(u,v) \]
- Example: \( z=x^2+y^2 \)
- \[ x(u,v) =u, \qquad y(u,v) =v,\qquad z(u,v)=u^2+v^2 \]
- \[ \begin{align} \mathbf{\vec{r}}(u, v) &= u\mathbf{\hat{i}}+v\mathbf{\hat{j}}+(u^2+v^2)\mathbf{\hat{k}}\\ &or\\ &= \langle u,v,u^2+v^2\rangle \end{align} \]
- Parameter domain \[ D=[-1,1]\times[-1,1] \]
- Parameter domain \[ D=\{(u,u)| \;u^2+v^2 \leq 1\} \]
Example: Cones
- \[ \mathbf{\vec{r}_1}(u, v)= \langle v, u, \sqrt{u^2+v^2}\rangle \]
Parameter domain \[ D=\{(u,v)|\; u^2+v^2\leq 1 \} \]
-
\[ \mathbf{\vec{r}_2}(u, v)= \langle v\cos u, v\sin u, v\rangle \]
Parameter domain \[ \begin{align} D&= \{(u,v)|\; 0 \leq u \leq 2\pi,\; 0\leq v\leq 1 \}\\ &=[0,2\pi]\times[0,1] \end{align} \]
\( \star \) Parameterizations are not unique \( \star \)
Example: Plane
- \( P(1,1,1), v_1=(-2,1,0), v_2=(2,-3,1) \)
Example: Sphere
- Representation in spherical coordinates \[ \begin{align} \rho&=\text{constant}\\ u&=\theta\\ v&=\phi \end{align} \]
- Then \[ \mathbf{\vec{r}}(u, v)= \rho\cos u \sin v\mathbf{\hat{i}}+\rho\sin u \sin v\mathbf{\hat{j}}+\rho \cos v\mathbf{\hat{k}} \]
Parameter domain \[ \begin{align} D&=\{(u, v)| \;0\leq u \leq 2\pi,\; 0\leq v \leq \pi\}\\ &=[0,2\pi]\times[0,\pi] \end{align} \]
Example: Cylinders
\[ \mathbf{\vec{r}}(u, v)= \langle 2 \cos u,2 \sin u, v\rangle\\ D=[0,2\pi]\times[-3,3] \]
\[ \mathbf{\vec{r}}(u, v)= \langle 2 \cos u,5 \sin u, v\rangle\\ D=[0,2\pi]\times[-4,4] \]
\[ \mathbf{\vec{r}}(u, v)= \langle u, v, v^2\rangle\\ D=[-2,2]\times[-2,2] \]
Example: Other surfaces
Spiral
Molusc shell
Torus/Donut
Möbius strip
Points on surface
Surface area
- Suppose that a surface \( S \) has a vector equation \[ \mathbf{\vec{r}}(u, v) = x(u,v)\mathbf{\hat{i}}+y(u,v)\mathbf{\hat{j}}+z(u,v)\mathbf{\hat{k}}\\ (u,v)\in D \]
- Slice:
Divide \( D \) into rectangles \( R_{ij} \)
the patch \( S_{ij} \) corresponds to \( R_{ij} \)
- Approximate
- The patch \( S_{ij} \) by the parallelogram determined by vectors
- The edges of the patch that meet at \( P_{ij} \) by the vectors \[ \mathbf{\vec{r}}^*_u\; \Delta u_i\\ \mathbf{\vec{r}}^*_v\; \Delta v_j \]
- Area of patch with cross product \[ A(S_{ij})\approx || \left(\mathbf{\vec{r}}^*_u\; \Delta u_i \right) \times \left(\mathbf{\vec{r}}^*_v\; \Delta v_j \right) || \]
Surface area cont'd
- Approximate
- Area of patch with cross product \[ A(S_{ij})\approx ||\mathbf{\vec{r}}^*_u \times\mathbf{\vec{r}}^*_v || \Delta u_i \Delta v_j \]
- Sum and take limit
- \[ A(S)\approx \sum_{i}\sum_j ||\mathbf{\vec{r}}^*_u \times\mathbf{\vec{r}}^*_v || \Delta u_i \Delta v_j \]
- \( \Delta u_i \rightarrow 0 \) and \( \Delta v_j \rightarrow 0 \)
Surface area \[ A(S)=\iint\limits_D ||\mathbf{\vec{r}}_u \times \mathbf{\vec{r}}_v||dA \] \( D \) is the parameter domain
Example: surface area
- Find the surface area of the part of the paraboloid \( z = f(x,y)= x^2 +y^2 \) that lies under the plane \( z=1 \)
- \[ A(S)=\iint\limits_D ||\mathbf{\vec{r}}_u \times \mathbf{\vec{r}}_v||dA \]
- Parameterization \[ \mathbf{\vec{r}}(u, v)= \langle u,v,f(u,v)\rangle\quad D=\{(u,v)|\; u^2+v^2\leq 1 \} \]
- Derive \[ \mathbf{\vec{r}}_u = \langle 1,0,\frac{\partial f}{\partial u} \rangle \]
- \[ \mathbf{\vec{r}}_v = \langle 0,1,\frac{\partial f}{\partial v} \rangle \]
- \[ \mathbf{\vec{r}}_u \times \mathbf{\vec{r}}_v=\left|\left(\begin{array}{ccc} \mathbf{\hat{{i}}} & \hat{{\mathbf{j}}} & \hat{{\mathbf{k}}}\\ 1 & 0 & \frac{\partial f}{\partial u}\\ 0 & 1 & \frac{\partial f}{\partial v}\\ \end{array}\right)\right| \]
- \[ \mathbf{\vec{r}}_u \times \mathbf{\vec{r}}_v=- \frac{\partial f}{\partial u} \mathbf{\hat{{i}}} + \frac{\partial f}{\partial v} \mathbf{\hat{{j}}}+1\mathbf{\hat{k}} \]
- \[ ||\mathbf{\vec{r}}_u \times \mathbf{\vec{r}}_v||=\sqrt{\left(\frac{\partial f}{\partial u} \right)^2+\left(\frac{\partial f}{\partial v} \right)^2 +1} \]
Example: surface area
- \[ ||\mathbf{\vec{r}}_u \times \mathbf{\vec{r}}_v||=\sqrt{\left(\frac{\partial f}{\partial u} \right)^2+\left(\frac{\partial f}{\partial v} \right)^2 +1} \]
- Since \( f(u,v)=u^2+v^2 \) \[ \frac{\partial f}{\partial u} = 2u\qquad \frac{\partial f}{\partial v}=2v \]
- Set up and evaluate \[ \begin{align} A(S) &= \iint\limits_D \sqrt{(2u)^2+(2v)^2+1}= \iint\limits_D \sqrt{4(u^2+v^2)+1}dA\end{align} \]
- Using polar coordinates \[ \begin{align} A(S) & = \int_0^{2\pi} \int_0^1 \sqrt{4r^2+1}\;rdrd\theta\end{align} \]
- \[ \begin{align} A(S)&= 2\pi \left( \frac{1}{8}\right)\left(\frac{2}{3}\right) \left(4r^2+1\right)^{3/2}\bigg\vert_0^1\\ &= \frac{\pi}{6}\left(5^{3/2} -1 \right)\end{align} \]
Tangent planes
Example: tangent planes